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Journal of Modern Optics

Schrödinger and Heisenberg formulation of classical pulse propagation with dispersion and attenuation

Leon W. Cohen

Published Oct 12, 2016• 1 citation
Publisher

Abstract

We consider the possibility of formulating classical wave prorogation in a quantum mechanical formalism. The classical wave may have dispersion and attenuation. We derive a Schrödinger type equation of motion where the equivalent Hamiltonian is the dispersion relation. We also derive a Heisenberg equation of motion for the cases with and without damping. When there is damping, we show that the Heisenberg equation of motion contains both a commutator and an anticommutator.

DOI
10.1080/09500340.2016.1242791

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We consider the possibility of formulating classical wave prorogation in a quantum mechanical formalism. The classical wave may have dispersion and attenuation. We derive a Schrödinger type equation of motion where the equivalent Hamiltonian is the dispersion relation. We also derive a Heisenberg equation of motion for the cases with and without damping. When there is damping, we show that the Heisenberg equation of motion contains both a commutator and an anticommutator.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrödinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.
We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations.
In this work, we propose an alternative formulation for the description of quantum dynamics based on a displacement interpretation of the Schrödinger equation. Startingfrom a generalized wave propagation framework, we show that the Schrödinger equation can be recovered in a form that admits a displacement-type interpretation ofthe wave function. Within this formulation, we derive a momentum-related relation, referred to as the Schrödinger momentum equation, which provides an effective description of quantumpropagation. This relation allows for a unified discussion of several foundational aspects of quantum mechanics, including interference phenomena, coherence anddecoherence, and quantum correlations. The proposed framework is deterministic and explicitly nonlocal, and remains consistent with established experimental observations. In addition, we examine theimplications of this formulation for the interpretation of the Heisenberg Uncertainty Principle, showing that uncertainty-type relations can be naturally obtained within thisapproach while preserving measurement-dependent limitations. Overall, the present work offers a complementary perspective on quantum mechanics that may contribute to ongoing discussions on its conceptual foundations.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.